PHYSICAL REVIEW B
VOLUME 55, NUMBER 19
15 MAY 1997-I
Comparative study of the stopping power of graphite and diamond
W. Käferböck and W Rössler
Universtiät Linz, Institut für Experimentalphysik, A-4040 Linz, Austria
V. Necas
Slovak Technical University, Department of Nuclear Physics and Technology, Bratislava, Slovakia
P. Bauer
Universtiät Linz, Institut für Experimentalphysik, A-4040 Linz, Austria
M. Peñalba
Departamento de Fı́sica Aplicada I, E.T.S.I.I y T., Universidad del Paı́s Vasco, Bilbao, Spain
E. Zarate and A. Arnau
Departamento de Fı́sica de Materiales, Facultad de Quı́mica, Universidad del Paı́s Vasco, San Sebastián, Spain
~Received 23 October 1996!
The stopping cross section of graphite and diamond for hydrogen and helium projectiles in the energy range
20<E<80 keV/amu has been measured. It is found that in the case of graphite the stopping cross section for
protons is up to 40% larger than that of diamond, while for He projectiles the difference is about one-half of
that. The results are explained using the charge-state approach to stopping, that includes in an approximate
manner the energy loss in charge-exchange processes and a simple model for the valence-electron excitations
of the two allotropic forms of carbon. @S0163-1829~97!07119-1#
I. INTRODUCTION
The stopping power for low atomic number materials and
light incident ions depends on the physical and chemical
state of the sample, especially when the velocity of the incident ion is about the Fermi velocity of the material. The
interaction of the incident ion with the solid can be divided
into two parts: one is due to the interaction with the valence
electrons, and the other is due to the interaction with the ion
cores. The first one plays an important role in the range of
velocities under consideration, and therefore the stopping
power depends strongly on the state of the sample. One of
the best-suited materials to study these effects is carbon,
which can be found in several allotropic forms: graphite,
amorphous carbon, diamond, and fullerenes. In this work we
present the experimentally measured and theoretically calculated stopping cross sections of diamond and graphite for
protons, as well as the ratio of the stopping cross sections for
a particles in a narrow energy range, in order to investigate
the allotropic effect.
The plasmon losses of diamond and graphite, well known
in literature, are found to be 35 and 27 eV, respectively.1 It
should be noted that these plasmon energies are not too far
from the values obtained in the free-electron-gas model from
the atomic densities n A of these materials, assuming a number of four electrons per atom contributing to the plasmon.
For a summary of the properties of different carbon phases,
see Table I. The values of the plasmon energies are rather
large, corresponding to rather high densities ~and low r S values, where r S is defined from the valence electron density
n el by r S 5 @ 3/4p n el# 1/3). An allotropic effect is clearly to be
expected, in terms of stopping power as well as in terms of
0163-1829/97/55~19!/13275~4!/$10.00
55
the stopping cross section. The allotropic effect in the stopping power is of the opposite direction compared to the stopping cross section. In the stopping power the density enters
twofold, via the number of collision partners and via the
strength of the interactions. The trivial density dependence
~i.e., the number of collision partners per unit length! is removed when looking at the cross section. Thus, in terms of
physics, the stopping cross section is more illuminative than
the stopping power.
II. EXPERIMENT
Similarly to our earlier measurements on oxides2 and on
amorphous carbon3 the stopping cross section was measured
by Rutherford backscattering ~RBS! comparing the spectrum
heights obtained from two targets for the same incident
charge. He1 and H1 ions of velocities around v 0 (5c/137)
are used as projectiles, highly oriented pyrolytic graphite
~HOPG! and polished natural diamond have been used as
target materials. The choice of targets of the same atomic
number simplifies the data evaluation, because the scattering
cross section and the kinematic factor are identical in both
TABLE I. Electronic and structural properties for the two carbon phases.
Mass density r (g/cm3)
Atomic density n A ~a.u.!
plasmon energy ~eV!
density parameter r S
13 275
Diamond
Graphite
3.516
2.6131022
35
1.224
2.25
1.6731022
27
1.447
© 1997 The American Physical Society
13 276
W. KÄFERBÖCK et al.
targets, and the spectrum heights are expected to differ only
because of different stopping cross sections. Both targets
were exposed to the same amount of primary ions, and the
stopping information was extracted from the spectrum
heights. Special care was taken to ensure reliability of the ion
current integration using the setup described earlier.4 Repeated test runs showed reproducibility and consistency of
the resulting ion charge values within 1%.
To extract the stopping information from the spectrum
heights is only possible in case that channeling in these materials is avoided. This is not trivial, because both diamond
and graphite have crystal directions in which channeling can
occur. In the HOPG used, to avoid channeling is simpler,
because here only perpendicular incidence and exit of the
projectiles will lead to channeling and blocking, respectively.
In diamond, proper crystal directions for which randomlike
spectra are obtained have to be searched carefully. In detailed angular scans those directions have been selected for
which the height of the high-energy edge of the measured
spectrum coincides within 3% with that expected for an
amorphous target as obtained from computer simulations,
analogously to our measurements on LiNbO3. 5 The reason to
choose natural diamond was that this material is quite pure
without any hydrogen contamination as in other types of
artificial diamonds.
From the heights of the RBS spectra of graphite, H gra ,
and of diamond, H dia the ratio of the stopping cross sections
in graphite S gra and diamond S dia follows from
H gra @ S dia# S dia~ E x !
5
5
,
H dia @ S gra# S gra~ E x !
~1!
where E x is a mean energy which may be determined following Ref. 6. We stress the point that, in this special case,
where the kinematic factor is identical for both targets, the
systematic errors of evaluating S dia /S gra from the previous
equation are negligible because they cancel each other to
high precision. The evaluation of the spectrum heights was
done as described in Ref. 2.
The uncertainty of the resulting ratios S dia /S gra is estimated to be 5%, containing contributions from current
integration4 ~61%!, the determination of the spectrum height
~63%!, and a possible residual influence of channeling in
diamond ~63%!. This estimate is in concordance with the
observed scatter of the experimental data. For hydrogen projectiles, we can use our earlier measurements of the stopping
ratio3 of graphite and amorphous carbon, S gra /S am , to obtain
absolute values S dia and S gra , but with a somewhat larger
uncertainty ~7%!. The measurements for H1 ions were done
using protons and deuterons which are known to have identical stopping properties at the same speed. In order to
present the results of both projectiles in the same figure, we
plot the stopping cross section for H1 ions as a function of
E/A, where E denotes the mean energy and A the number of
nucleons. For He ions, only 4 He1 projectiles have been
used, therefore in this case the mean energy of 4 He1 projectiles is used as abscissa.
III. THEORETICAL CALCULATIONS
The energy-loss calculation in solids is a complicated
problem due to its many-body character, and especially in
55
the energy region where the maximum of the stopping is
located ( v >Z 2/3
1 v 0 ). In this region there is a succession of
electron capture and loss events by the incident ion which
complicates the treatment. As a consequence the charge state
of the incident ion beam is modified. If the thickness of the
sample is larger than a minimum value an equilibrium charge
state distribution independent of the initial state of the beam
is reached. The population of the different charge states can
be calculated from the capture and loss rates ~probability per
unit time of having a capture or a loss event respectively,
G C,L !. Associated with the charge changing processes there
is an additional energy loss that can be calculated in a similar
way as the capture and loss probabilities.7 It typically
amounts for 10–20 % of the total losses as long as Z 1
!Z 2 :
S D
dE
dR
5
C,L
1
v
E
dv v
dG C,L
,
dv
~2!
where v is the electron transition energy ~atomic units are
used for the theoretical expressions!. In the energy range that
we consider, hydrogen projectiles can be found in two different charge states: H0 and H1. So the energy loss can be
written as a weighted sum
F
G F G
F S D S DG
dE
dE 0
dE 1
5F 0
~ H ! 1F 1
~H !
dR
dR
dR
1 F0
dE
dR
1F 1
L
dE
dR
.
~3!
C
The total energy loss is the addition of the energy losses
associated with each charge state ~weighted by its corresponding fraction F 0,1 !, plus the one due to the capture and
loss processes. For He projectiles, He0 and He1 are to be
used instead of H0 and H1. The He energy loss, above 100
keV, is mostly due to target electron excitations by He1 ions.
We use an electron-gas model for the target to represent
the valence-electron excitations of carbon with two different
densities for diamond and graphite, respectively ~see Table
I!. Strictly speaking, the electrons are not free, but as long as
the gap energy is small compared to the plasmon energy and
to the mean transition energies (>m vv F ), we regard the
valence electrons as nearly free.
The energy loss of the He1 and H1 ions is calculated in
the dielectric formalism using a random-phaseapproximation ~RPA! dielectric function with the corresponding density parameter for each material. This treatment
includes both electron-hole pair and plasmon excitations in a
self-consistent way.8 The dielectric approach can be used as
long as only the H1 fraction is relevant, i.e., at high velocities. In the case of He1 ions we also account for the screening by the electron bound to the projectile.9
We have calculated dE/dR (H0) by independently obtaining contributions from the electron-hole pairs and those
from the plasmon excitations. The former is obtained using a
binary encounter approximation.7 The plasmon contribution,
although small, is obtained in linear theory using the RPA
dielectric function, taking into account that nonlinear effects
are only important at low ion velocities. In order to calculate
COMPARATIVE STUDY OF THE STOPPING POWER OF . . .
55
FIG. 1. Stopping cross section ~in 10215 eV cm22! of diamond
for protons. The curve labeled total is the result of the theoretical
calculation, while the square points are experimental data. H1 and
H0 refer to the contributions from H1 and H0 to the total stopping.
C&L is the contribution from capture and loss processes.
the equilibrium charge-state distribution for protons we have
considered two kinds of processes10 that can lead to a
charge-exchange event.
~a! Auger processes. An electron can be captured or lost
by the incident ion due to the interaction with the valenceband electrons, inducing at the same time an excitation in the
band ~electron-hole pairs or plasmons!.
~b! Resonant processes which are due to the interaction of
the ions with the lattice. The interaction potential is seen by
the incident ion as a time-dependent periodic potential that
can induce transitions between the bound states of the ion
and the continuum states.
Those processes where inner shell electrons of the lattice
ions can be captured by the incident ion are not important at
these energies, as the carbon K-shell electrons are bound by
284 eV.
In the case of diamond we calculated the probabilities for
the capture and loss events by the previously described
mechanisms. The response of the valence band is represented
by a RPA dielectric function with the appropriate r S parameter ~Table I!. The ions are organized in a face-centeredcubic lattice with a basis, and we represent the interaction
potential by a statically screened ~Thomas-Fermi! Hartree
one,
V~ q !5
4p
q
2
F
2
1k TF
Z 22
G
2
,
„11 ~ q/2a k ! 2 …2
13 277
FIG. 2. Same as Fig. 1 for graphite instead of diamond.
s L5
F1
s
F0 C
where s C > s AC .
~5!
The stopping cross section associated with the capture and
loss processes, S C,L , can be obtained by multiplying the
cross section, s C,L , by a mean transition energy for the capture and loss events, respectively,12
S D
~6!
S L 5 s L DE L 5 s L E b ,
~7!
S C 5 s C DE C 5 s C
v2
,
2
where E b is the binding energy of the electron lost by the
incident ion.
IV. RESULTS
In Fig. 1 we show the stopping cross section for protons
in diamond as a function of the incident ion energy. The dots
represent experimental results, and the curve labeled TOTAL
is the theoretical calculation. The curves H0, H1, and C&L
represent each partial contribution to the stopping cross section. The agreement is rather good except in the lowest energy region. In the low-velocity region the H0 contribution
~4!
where Z 2 56, a k 55.7, and k TF is the Thomas-Fermi screening wave vector.
For the graphite we used charge-state fraction data available in the literature.11 We know from calculations in some
other materials and in diamond that in the case of protons at
these energies the resonant capture probabilities are much
smaller than the Auger ones; thus we approximate s C 5 s AC
1 s RC > s AC . We calculated the Auger process cross sections
using the corresponding electron-gas density ~Table I!. From
the experimental charge-state fractions one can deduce the
cross section for the loss events:
FIG. 3. Ratio of the stopping cross sections of graphite and
diamond for protons.
W. KÄFERBÖCK et al.
13 278
55
is the calculated one for He1 ions, while the different symbols correspond to the experimental data in the energy range
80 keV,E,280 keV. There is a fair agreement between the
theory and the experimental data.
V. CONCLUSIONS
FIG. 4. Same as Fig. 3 for He projectiles.
dominates the stopping cross section while it becomes negligible in the high velocity range which is dominated by the
H1 contribution. The C&L cross section has its maximum
value at velocities close to the Fermi velocity (m p v F 2 /2
>60 keV!, where it is about a 10% of the total stopping
cross section.
Figure 2 is equivalent to the previous one, but it shows the
values for graphite. We again find a worse agreement in the
low-energy range. The behavior of the C&L curve is not so
reliable at low energies, which is partially due to the fact that
we have used approximate mean transition energies to calculate the capture and loss contribution.
In Fig. 3 the theoretical and experimental results for the
ratio of the graphite and for diamond stopping cross sections
are represented together. The picture shows clearly the existence of a noticeable allotropic effect in the stopping cross
section. The stopping ratio observed comparing the results
for the two materials is about 25–30 %, and is very well
reproduced by the theoretical results, even though there is a
discrepancy between experimental and theoretical values for
each material.
In Fig. 4 we show the ratio of the stopping cross sections
of diamond and graphite for He ions. The continuous curve
1
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The different stopping cross sections of graphite and diamond for protons have been successfully explained using the
charge-state approach.6 Although we typically overestimate
the values of both the two cross sections and find a larger
discrepancy at the lower velocities, we consider that the
agreement between theory and experiment is quite satisfactory considering the simplicity of the model, particularly the
way of treating the energy loss in charge-changing processes.
It is worth mentioning that the ratio of the atomic densities of
diamond and graphite is 1.56, which shows that the protons
are more efficiently slowed down in the case of diamond, as
the measured difference in stopping cross section is below
50%. In conclusion, although diamond has a band gap in its
electronic excitation spectrum ~>4 eV! and graphite has a
much smaller one ~>1 eV!, these gaps are not relevant as far
as electronic stopping is concerned in this energy range (E
.20 keV!, and the energy loss of hydrogen ions may be
interpreted in terms of a free-electron-gas description.13
ACKNOWLEDGMENTS
We gratefully acknowledge partial financial support by
the HCM program of the EC under Contract No.
ERBCHRXCT94079. V.N. wants to express his gratitude for
financial support by the Austrian Bundesministerium für
Wissenschaft und Forschung, and wants to thank cordially
Professor H. Paul for supporting this cooperation and for the
kind hospitality of the Abterilung für Atom and Kernphysik
at the JKU in Linz. P.B. acknowledges Walter Mican for
supplying the diamond samples. E.Z. is grateful for partial
financial support by Iberdrola S.A.
8
J. Lindhard, K. Dan. Vidensk. Selks. Mat. Fys. Medd. 28, 8
~1954!.
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T. L. Ferrel and R. H. Ritchie, Phys. Rev. B 16, 115 ~1977!.
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Physics, edited by H. Ehrenreich and D. Turnbull ~Academic,
New York, 1990!, Vol. 43, p. 230.
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Fainstain, and P. M. Echenique, Phys. Rev. B 49, 6470 ~1994!.
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P. M. Echenique, F. J. Garcı́a de Abajo, V. H. Ponce, and M. E.
Uranga, Nucl. Instrum. Methods Phys. Res. Sect. B 96, 583
~1995!.